sstt st rtr - PowerPoint PPT Presentation

❈♦♥s✐st❡♥t ♠♦❞❡❧ s❡❧❡❝t✐♦♥ ❝r✐t❡r✐❛ ❛♥❞ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡st ❢♦r ❝♦♠♠♦♥ t✐♠❡ s❡r✐❡s ♠♦❞❡❧s ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❑✳ ❑❛r❡ ✭P❛r✐s ✶✮ ❛♥❞ ❲✳ ❑❡♥❣♥❡ ✭❈❡r❣②✮ ❏❡❛♥✲▼❛r❝ ❇❛r❞❡t✱ ❙❆▼▼✱ ❯♥✐✈❡rs✐té P❛r✐s ✶✱ ❋r❛♥❝❡ ❜❛r❞❡t❅✉♥✐✈✲♣❛r✐s✶✳❢r ▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡ ✷ ❏✉♥❡ ✷✵✷✵ ❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✶ ✴ ✸✷

✶ ❆♥ ❡①❛♠♣❧❡ ✷ ❈❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✸ ●❛✉ss✐❛♥ ◗✉❛s✐✲▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞ ❊st✐♠❛t✐♦♥ ✹ ❈♦♥s✐st❡♥❝② ♦❢ ❛ ♣❡♥❛❧✐③❡❞ ◗▼▲ ❝r✐t❡r✐♦♥ ❛♥❞ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡st ✺ ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❖✉t❧✐♥❡ ✶ ❆♥ ❡①❛♠♣❧❡ ✷ ❈❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✸ ●❛✉ss✐❛♥ ◗✉❛s✐✲▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞ ❊st✐♠❛t✐♦♥ ✹ ❈♦♥s✐st❡♥❝② ♦❢ ❛ ♣❡♥❛❧✐③❡❞ ◗▼▲ ❝r✐t❡r✐♦♥ ❛♥❞ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡st ✺ ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✷ ✴ ✸✷

❊①❛♠♣❧❡ ❖❜s❡r✈❡ t❤❡ tr❛❥❡❝t♦r② ♦❢ t❤❡ ❧♦❣❛r✐t❤♠✐❝ r❡t✉r♥s ♦❢ ❙✫P ✺✵✵ ✿ 0.04 0.02 logSP500 0.00 −0.02 −0.04 2012 2013 2014 2015 2016 2017 Date = ⇒ ❚✇♦ ❛✐♠s ✿ ❈❤♦s❡ ❛♥ ✧♦♣t✐♠❛❧✧ ♠♦❞❡❧ ❢♦r t❤❡s❡ ❞❛t❛ ❀ ❚❡st ✐ts ❣♦♦❞♥❡ss✲♦❢✲✜t✳ ❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✸ ✴ ✸✷

❚✇♦ ✐♥t✉✐t✐✈❡ ❞❡✜♥✐t✐♦♥s ▲❡t ( X t ) t ∈ Z ❜❡ ❛ t✐♠❡ s❡r✐❡s ✭s❡q✉❡♥❝❡ ♦❢ r✳✈✳ ♦♥ (Ω , A , I P) ✮ N ∗ ✱ ∀ ( t ✶ , . . . , t k ) ∈ Z k ✱ ( X t ) t ∈ Z ✐s ❛ st❛t✐♦♥❛r② ♣r♦❝❡ss ✐❢ ∀ k ∈ I � � L � � ∼ ❢♦r ❛❧❧ h ∈ Z ✳ X t ✶ , . . . , X t k X t ✶ + h , . . . , X t k + h ❆ss✉♠❡ t❤❛t ( ξ t ) t ∈ Z ✐s ❛ ✇❤✐t❡ ♥♦✐s❡ ✭❝❡♥t❡r❡❞ ✐✳✐✳❞✳r✳✈✳✮ � � N → I R I ( X t ) t ∈ Z ❝❛✉s❛❧ ♣r♦❝❡ss ✐❢ ∃ H : I R s✉❝❤ ❛s X t = H ( ξ t − k ) k ≥ ✵ ✳ ❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✹ ✴ ✸✷

❆❘▼❆ ♣r♦❝❡ss❡s ✭❲❤✐tt❧❡✱ ✶✾✺✶✮ R p ✱ ( b j ) ∈ I R q ❲✐t❤ ( ξ t ) t ∈ Z ❛ ✇❤✐t❡ ♥♦✐s❡✱ ( a i ) ∈ I ❆❘▼❆✭ p , q ✮ ♣r♦❝❡ss ✿ ✇✐t❤ a p � = ✵ ❛♥❞ b q � = ✵✱ ❢♦r ❛♥② t ∈ Z X t + a ✶ X t − ✶ + · · · + a p X t − p = ξ t + b ✶ ξ t − ✶ + · · · + b q ξ t − q ❙t❛t✐♦♥❛r✐t② ❛♥❞ ❝❛✉s❛❧✐t② ✿ ✶ + a ✶ z + · · · + a p z p � = ✵ ❢♦r ❛♥② | z | ≤ ✶✳ 10 5 0 −5 −10 −15 0 200 400 600 800 1000 ❆❘▼❆✭✶ , ✶✮ ❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✺ ✴ ✸✷

●❆❘❈❍ ♣r♦❝❡ss❡s ✭❊♥❣❡❧✱ ✶✾✽✷✮ ✭❇♦❧❧❡rs❡✈✱ ✶✾✽✻✮ R p R q ❲✐t❤ ( ξ t ) t ∈ Z ❛ ✇❤✐t❡ ♥♦✐s❡✱ ( c i ) ∈ I + ✱ ( d j ) ∈ I + ●❆❘❈❍✭ p , q ✮ ♣r♦❝❡ss ✿ ✇✐t❤ c ✵ , c p > ✵ ❛♥❞ d q > ✵✱ ❢♦r ❛♥② t ∈ Z � X t = σ t ξ t , σ ✷ c ✵ + c ✶ X ✷ t − ✶ + · · · + c p X ✷ t − p + d ✶ σ ✷ t − ✶ + · · · + d q σ ✷ = t t − q � q ✵ ) � p E( ξ ✷ j = ✶ d j + I i = ✶ c i < ✶ = ⇒ ❙t❛t✐♦♥❛r✐t② ❛♥❞ ❝❛✉s❛❧✐t② 30 20 10 0 −10 −20 −30 0 200 400 600 800 1000 ●❆❘❈❍✭✶ , ✶✮ ❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✻ ✴ ✸✷

▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❛♥❞ ●♦♦❞♥❡ss✲♦❢✲✜t t❡st ❈♦♥s✐❞❡r ❛ ❢❛♠✐❧② M ♦❢ ♠♦❞❡❧s✳ ❋♦r ✐♥st❛♥❝❡✱ � ❆❘▼❆ ( p , q ) ♦r ●❆❘❈❍ ( p ′ , q ′ ) , M = � ✇✐t❤ ✵ ≤ p , p ′ ≤ p max , ✵ ≤ q , q ′ ≤ q max ❲❡ ✇❛♥t t♦ ✿ ❈❤♦s❡ ❛♥ ✧♦♣t✐♠❛❧✧ ♠♦❞❡❧ ✐♥ M ❢♦r ( X ✶ , . . . , X n ) ❀ ❊st✐♠❛t❡ ✐ts ♣❛r❛♠❡t❡rs ❀ ❚❡st ✐ts ❣♦♦❞♥❡ss✲♦❢✲✜t✳ ❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✼ ✴ ✸✷

❖✉t❧✐♥❡ ✶ ❆♥ ❡①❛♠♣❧❡ ✷ ❈❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✸ ●❛✉ss✐❛♥ ◗✉❛s✐✲▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞ ❊st✐♠❛t✐♦♥ ✹ ❈♦♥s✐st❡♥❝② ♦❢ ❛ ♣❡♥❛❧✐③❡❞ ◗▼▲ ❝r✐t❡r✐♦♥ ❛♥❞ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡st ✺ ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✽ ✴ ✸✷

❊①❛♠♣❧❡s ✿ ❈❛✉s❛❧ ❆❘❬ ∞ ❪ ❛♥❞ ❆❘❈❍✭ ∞ ✮ ♠♦❞❡❧s ❲✐t❤ ( ξ t ) t ∈ Z ❛ ✇❤✐t❡ ♥♦✐s❡✱ � ∞ ❆❘ ( ∞ ) ♣r♦❝❡ss❡s X t = θ i X t − i + ξ t i = ✶ p q � � = ⇒ ❈❛✉s❛❧ ❆❘▼❆✭ p , q ✮ ♣r♦❝❡ss❡s X t + a i X t − i = ξ t + b i ξ t − i ✳ i = ✶ i = ✶ ❆❘❈❍✭ ∞ ✮ ♣r♦❝❡ss❡s✱ ✭ ❘♦❜✐♥s♦♥✱ ✶✾✾✶ ✮✱ ✇✐t❤ b ✵ > ✵ ❛♥❞ b j ≥ ✵ � X t = σ t ξ t , φ ✵ + � ∞ σ ✷ j = ✶ φ j X ✷ = t − j . t ⇒ ●❆❘❈❍✭ p , q ✮ ♣r♦❝❡ss❡s✱ ✇✐t❤ c ✵ > ✵✱ c j , d j ≥ ✵✱ c p , d q > ✵ = � X t = σ t ξ t , c ✵ + � p t − j + � q σ ✷ j = ✶ c j X ✷ j = ✶ d j σ ✷ = t t − j ❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✾ ✴ ✸✷

❆ ❝♦♠♠♦♥ ❢r❛♠❡ ❢♦r st✉❞②✐♥❣ t✐♠❡ s❡r✐❡s ❆ ❝♦♠♠♦♥ ❝❧❛ss ♦❢ ♠♦❞❡❧s ❢♦r ❆❘✱ ❆❘▼❆✱ ❆❘❈❍ ❛♥❞ ●❆❘❈❍ ♣r♦❝❡ss❡s ✿ ❈❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✿ ❝❧❛ss CA ( M , f ) X t = M ( X t − ✶ , X t − ✷ , . . . ) ξ t + f ( X t − ✶ , X t − ✷ , . . . ) , ∀ t ∈ Z , ❛✳s✳ . N ❀ R I M ( · ) ❛♥❞ f ( · ) ❛r❡ r❡❛❧ ✈❛❧✉❡❞ ❢✉♥❝t✐♦♥ ♦♥ I � | ξ ✵ | r � ( ξ t ) t ∈ Z ❛ ✇❤✐t❡ ♥♦✐s❡ ✇✐t❤ I E( ξ ✵ ) = ✵ ❛♥❞ I E < ∞ ✱ r ≥ ✶✳ ❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✶✵ ✴ ✸✷

❊①t❡♥s✐♦♥s ♦❢ ✉♥✐✈❛r✐❛t❡ ❆❘❈❍ ♠♦❞❡❧s ❚●❆❘❈❍✭ ∞ ✮ ♣r♦❝❡ss❡s✱ ✭ ❩❛❦♦ï❛♥✱ ✶✾✾✹ ✮✱ ✇✐t❤ b ✵ , b + j , b − j ≥ ✵  X t = σ t ξ t ,  � � ✳ � ∞ b + j max( X t − j , ✵ ) − b − σ t = b ✵ + j min( X t − j , ✵ )  j = ✶ ❆P❆❘❈❍✭ δ, p , q ✮ ♣r♦❝❡ss❡s✱ ✭ ❉✐♥❣ ❡t ❛❧✳ ✱ ✶✾✾✸ ✮  X t = σ t ζ t ,  � p α i ( | X t − i | − γ i X t − i ) δ + � q σ δ j = ✶ β j σ δ = ω + t − j ,  t j = ✶ ✇✐t❤ δ ≥ ✶✱ ω > ✵✱ − ✶ < γ i < ✶ ❛♥❞ α i , β j ≥ ✵✳ ❏✳✲▼✳ ❇❛r❞❡t✱ P❛r✐s ✶ ✭▼▼▼❙✷ ❈■❘▼ ❈♦♥❢❡r❡♥❝❡✮ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ❢♦r ❝❛✉s❛❧ ❛✣♥❡ ♠♦❞❡❧s ✶✶ ✴ ✸✷